Expression For The Excess Pressure Due To Water Hammer
When a valve with a volumetric flow rate Q is closed, an excess pressure δP is created upstream of the valve, whose value is given by the Joukowsky equation:
In this expression:
- overpressurization δP is expressed in Pa;
- Q is the volumetric flow in m3/s;
- Zh is the hydraulic impedance, expressed in kg/m4/s.
The hydraulic impedance Zh of the pipeline determines the magnitude of the water hammer pulse. It is itself defined by:
with:
- ρ the density of the liquid, expressed in kg/m3;
- A cross sectional area of the pipe, m2;
- Beff effective modulus of compressibility of the liquid in the pipe, expressed in Pa.
The latter follows from a series of hydraulic concepts:
- compressibility of the liquid, defined by its adiabatic compressibility modulus Bl, resulting from the equation of state of the liquid generally available from thermodynamic tables;
- the elasticity of the walls of the pipe, which defines a modulus of equivalent compressibility Beq. In the case of a pipe of circular cross section whose thickness e is small compared to the diameter D, the equivalent modulus of compressibility is given by the following formula: ; in which E is the Young's modulus (in Pa) of the material of the pipe;
- possibly compressibility Bg of gas dissolved in the liquid, defined by:
- γ being the ratio of specific heats of the gas
- α the rate of ventilation (the volume fraction of undissolved gas)
- and P the pressure (in Pa).
Thus, the effective compressibility modulus is:
As a result, we see that we can reduce the water hammer by:
- increasing the pipe diameter at constant flow, which reduces the inertia of the liquid column;
- choosing to use a material with a reduced Young's modulus;
- introducing a device that increases the flexibility of the entire hydraulic system, such as a hydraulic accumulator;
- where possible, increasing the percentage of undissolved air in the liquid.
Read more about this topic: Water Hammer
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